LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 41

The continuum Euler-Arnold equations for the motion of an N-dimensional rigid body

are given by

^ £ - [ M , f t ] . , (2.142)

where Q £ o(N) is defined through

M= JQ + QJ . (2.143)

These equations are the continuum limit of the discrete Euler-Arnold equation (1.12) (see

[MV]) and, as noted above, they have the same integrals. Question: is equation (1.12) the

time-r map for (2.142) for some r 0?

The answer is "no" by the following argument. The interpolating flow for (1.12) is

^ = [(w_ log M(tt •) )(0), M] . (2.124)

Evaluating (TT- log M(t, •) )(0) for small M, we find after a straightforward calculation that

[(7r-logM(t,.)(0),A/] = -[ft,M] + K(M) + 0(M6) , (2.144)

where

K{M)

v fir ( f r 1 ,** l x3 w i i ds \ A" dX ( 2 - 1 4 5 )

T-ttJ-irJc s- \_xi s- \*xi 2 T T Z

/

( 1 -

A2)3

2m

and so (2.124) becomes

^ = [ M , f i ] + / « M ) + 0 ( M 6 ) ,

which agrees with (2.142) to third order. However a simple computation shows that K(M)

is not identically zero for all J and M, and hence (2.142) cannot interpolate (1.12) for any

r 0.

As a final remark we note that in the continuum limit, M = em(et), e | 0, equation

(2.124) reduces in scaled time T = et to the equation

— ( m -f

fiJ2)

= [m +

f.iJ2,

Q, -f fiJ] ,

m = Jft-f ftJ ,

which is Manakov's Lax-pair form for the Euler-Arnold equations (see [Man]).